Jie Yuan, Meixiong Chen, Yingying Li, Zhongqi Tan, and Zhiguo Wang, "Reanalysis of generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators," Opt. Express 21, 2297-2306 (2013)
By utilizing the novel coordinate system for Gaussian beam reflection and the generalized ray matrix for spherical mirror reflection, the generalized sensitivity factors SD1, ST1, SD2 and ST2 influenced by both the radial and axial displacements of a spherical mirror in a nonplanar ring resonator have been obtained. Besides, the singular points of different kinds of non-planar ring resonators under the conditions of incident angle A ranging from 0° to 45° or total coordinate rotation angle ρ ranging from 0°to 360°have also been obtained through the analysis of the determinant of the coefficient matrix of the linear equations. The analysis in this paper is important to the cavity design of non-planar ring resonators and it could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in non-planar ring resonators.
Fig. 1 Geometrical construction of a four-equal-sided non-planar ring resonator (NPRO), mj(j = 1,2,3,4):reflecting mirror with radius of Rj(j = 1,2,3,4), Pj(j = 1,2,3,4): terminal points of the resonator.
Fig. 2 Coordinate systems and corresponding coordinate rotations based on traditional coordinate system for Gaussian beam reflection (TCS) and novel coordinate system for Gaussian beam reflection (NCS) in four equal-sided non-planar ring resonators (NPRO), β: folding angle, m1 and m2: spherical mirrors with radius of R1 and R2, m3 and m4: planar mirrors, Aj(j = 1,2,3,4): incident angles on four mirrors, Pj(j = 1,2,3,4): terminal points of the resonator, Pe, Pf, Pg, Ph, O1, O2: the midpoints of straight lines P1P2, P2P3, P3P4, P4P1, P1P3 and P2P4 separately, φtj(j = 1,2,3,4) and φj(j = 1,2,3,4): coordinate rotation angles based on TCS and NCS respectively, nj(j = 1,2,3,4): the binormals at points Pj(j = 1,2,3,4), (xtj, yj, zj) and (xj, yj, zj)(j = 1,2,3,4): coordinate systems for the incident beam (based on TCS and NCS respectively) before being reflected from points Pj(j = 1,2,3,4), (xtjr, yjr, zjr) and (xjr, yjr, zjr)(j = 1,2,3,4): coordinate systems for the reflected beam (based on TCS and NCS respectively) after being reflected from points Pj(j = 1,2,3,4), δjz(j = 1,2,3,4): axial displacement of mirrors mj(j = 1, 2, 3, 4), δjx, δjy(j = 1,2): radial displacements of the spherical mirrors m1 and m2. (Note: The positive directions of yj and yjr(j = 1,2,3,4) are along the directions of nj(j = 1,2,3,4); the positive directions of zj and zjr(j = 1,2,3,4,b,c) are along the direction of beam propagation; (xt1, xt1r, x1, x1r), (xt2, xt2r, x2, x2r), (xt3, xt3r, x3, x3r) and (xt4, xt4r, x4, x4r) are located at the incident planes of P4P1P2, P1P2P3, P2P3P4 and P3P4P1 separately; the positive directions of δ1x, δ2x, δ1z, δ2z, δ3z and δ4z are along the directions of straight lines P2P4, P1P3, P1O2, P2O1, P3O2 and P4O1 separately; the positive direction of δjy (j = 1,2) is along the direction of nj(j = 1,2).)
Fig. 7 Stability map of NPRO and the track of the singular points under the condition of (a) ρ ranging from 0°to 360°and L/R ranging from 0 to 2, (b) ρ ranging from 0°to 360°and L/R ranging from 2 to 8, (c) A ranging from 0°to 45°and L/R ranging from 0 to 2, (d) A ranging from 0°to 45°and L/R ranging from 2 to 8. (Note: the stable and unstable regions are separated with solid lines; the tracks of the singular points are illustrated with the red marked lines)